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\textbf{Expression of the exponential envelope curve}
\begin{equation}
F_{\mathrm{env}}(D) = 
\begin{cases}
(F_0+R_1 K_0 D)\left[ 1-\exp(\frac{-K_0 D}{F_0}) \right], & 0\leq D \leq D_c \\ 
F_c - K_d (D-D_c), & D_c < D \leq D_u \\ 
0, & D > D_u
\end{cases}
\end{equation}

\textbf{Expression of the force intercept of the pinching line}
\begin{equation}
F_{\mathrm{intercept}} = 
\begin{cases}
\left(\frac{D_{un}}{D_y}\right) F_{I},  & \lvert D_{un} \rvert \leq D_y  \ \mathrm{and} \  \lvert D_{m,s}\rvert \leq D_y \\ 
F_{I},  & \lvert D_{un} \rvert \leq D_y \ \mathrm{and} \lvert D_{m,s} \rvert > D_y \\ 
F_{I} + \eta(F_{un}-F_y), & \lvert D_{un} \rvert > D_y 
\end{cases}
\end{equation}
where $(D_{un}, F_{un})$ are the coordinates of the unloading point, and $D_{m,s}$ is the maximum displacement on the same side.

\textbf{Expressions of the target displacement of the reloading curve}
\begin{equation}
    D_{\mathrm{tar}} = \beta\gamma^{\lambda}D_{m,o}
\end{equation}

\begin{equation}
\lambda = \frac{\Sigma E_{p,i} + \Sigma E_i}{E_f + \Sigma E_i}
\end{equation}
where $E_{p,i}$ is the energy dissipated in a primary half-cycle, $E_i$ is the energy dissipated in the follower half-cycles, and $E_f$ is the energy dissipated in a monotonic test to failure. 
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\textbf{Expressions of stiffness degradation}
\begin{equation}
K_{\mathrm{pinching}} = \begin{cases}
K_{p}, & \alpha \geq 0 \ \mathrm{and} \ D_m \leq D_y \\ 
K_{p}\left(\frac{D_y}{D_m}\right)^{\alpha_p}, & \alpha \geq 0 \ \mathrm{and} \ D_m > D_y \\ 
K_{p}\left(\frac{F_{un}}{D_{un}K_{0,s}}\right)^{\lvert\alpha_p\rvert}, & \alpha < 0
\end{cases}
\end{equation}

\begin{equation}
K_{\mathrm{unloading}} = \begin{cases}
R_uK_{0,s}, & \alpha_u \geq 0 \ \mathrm{and} \ \lvert D_{m,s}\rvert \leq D_y \\
R_uK_{0,s} \left(\frac{D_y}{\lvert D_{m,s}\rvert}\right)^{\alpha_u}, & \alpha_u \geq 0 \ \mathrm{and} \ \lvert D_{m,s}\rvert \leq D_y \\
R_u \left(\frac{F_{un}}{D_{un}K_{0,s}}\right)^{\lvert\alpha_u\rvert}, & \alpha_u < 0
\end{cases}
\end{equation}

\begin{equation}
K_{\mathrm{reloading}} = \begin{cases}
K_{0,o}, & \alpha_r \geq 0 \ \mathrm{and} \ \lvert D_{m,o}\rvert \leq D_y \\
K_{0,o} \left(\frac{D_y}{\lvert D_{m,o}\rvert}\right)^{\alpha_u}, & \alpha_u \geq 0 \ \mathrm{and} \ \lvert D_{m,o}\rvert \leq D_y \\
\left(\frac{F_{un}}{D_{un}}\right)^{\lvert\alpha_r\rvert}, & \alpha_r < 0
\end{cases}
\end{equation}
where $K_{0,s}$ and $K_{0,o}$ are the stiffness on the same and opposite side of the unloading point, respectively. $D_{m,s}$ and $D_{m,o}$ are the maximum or minimum displacement on the same and opposite side of the unloading point, respectively. 
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